How can we communicate our thinking?

The math teachers I work with often express frustration with their learners' inability to communicate their thinking when it comes to solving problems. If learners are stuck, they often struggle to articulate what they have tried. If the problem has been solved, learners have trouble explaining their efforts. Too often, learners respond to a teacher's question about their thinking with, "I don't know." This does not present teachers with the assessment data necessary to evaluate what learners can do or are trying to do, which makes it difficult to plan what comes next.


I encountered this same problem while working with fifth-graders on fraction computation. They were practiced in giving answers and even showing work but when I asked them to share their thinking they often said, "I don't know." This provided me with an opportunity to try a response suggested by Ellin Oliver Keene in a session at the 2009 MRA Conference: "Pretend that you did know - what would you say?" The fifth-graders found that this framing supported their communication efforts by freeing them to take a risk because they were "pretending."


While this got them talking about their thinking, they still needed more support to organize their efforts. I thought it would help to demonstrate what a reasoning recount might look like. I was introduced to the recount text form through Margaret Mooney's book, Text Forms and Features. Here is the model reasoning recount I wrote based on the prior efforts of the class to think about adding fractions.

Recently, I have been collaborating with Jennifer Brokofsky via Twitter and email about ways to connect reading, math, and writing. The figure below represents our current thinking.  I hope this vignette further demonstrates the link. We recognize that this is work in progress, and your support would be appreciated. Please share your thinking in the comments.

Are they related?

The following mathematics activity was originally written for middle school learners. It was intended to introduce  this  seventh grade, Michigan Grade Level Content Expectation: "Create and interpret scatter plots and find line of best fit; use an estimated line of best fit to answer questions about the data." While lessons on this expectation typically include middle schoolers measuring things like their own arm span and height to see if they are related, my experiences with adolescents suggest that they are often self-conscious about their bodies and uncomfortable with such activities. Instead, I came up with the idea for them to measure dolls and action figures to see if certain relationships fit what we know about the human body. Here is the activity using the Connected Math Project's Launch-Explore-Summarize learning plan format.
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Needs: One hour; learners in groups; The Da Vinci Code; The Vitruvian Man handout; something that measures distances; set of dolls or action figures (preferably one for every pair of students); and a method for collecting and graphing data


Launch: The Divine Proportion
Read "The PHI Section" from Dan Brown's The Da Vinci Code. Here is a paragraph to give you a taste:

Despite PHI’s seemingly mystical mathematical origins, Langdon explained, the truly mind-boggling aspect of PHI was its role as a fundamental building block in nature. Plants, animals, and even human beings all possessed dimensional properties that adhered with eerie exactitude to the ratio of PHI to 1. 

Test out the theory on The Vitruvian Man (modified for seventh graders)

Explore Data from Dolls
Assign each expert group to explore a relationship described in "The PHI Section"

• Total height to belly button height
• Distance of shoulder-fingertip to elbow-fingertip
• Distance of hip-floor to knee-floor

Here is the data collected from one set of dolls using Fathom:

Summarize Jigsaw
New groups will be formed that include one person from each expert group. Learners will share what they found during their exploration. Teacher walks around gathering information to summarize the activity with the whole group.
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In the Common Core State Standards, this topic is found in the high school statistics section:
Summarize, represent, and interpret data on two categorical and quantitative variables
6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

Do you see what I see?

We live right next to the bike path that runs around Spring Lake. The other day I witnessed the following scene that gave me a laugh. A parent and child were riding on a tandem bike (that I have since learned is sometimes called a tag-along). The funny part was that the parent was exerting a great deal of effort to keep the bike moving while the child had his head on the handlebars and his feet off the pedals. I did not have my camera handy, so I took a mental picture and later drew the following:

Of course, like any snapshot, it is difficult to tell what happened before that led to the scene or what happened after. But as almost always happens, it got me to thinking about education. I saw the picture as a metaphor for what seems to happen this time of year in a lot of classrooms. Before I go on, I wonder if you see what I see.

When does it work?

I have been sharing my experience teaching a group of fifth-graders how to problem-solve around fraction computation and using it as an opportunity to demonstrate the Teaching-Learning Cycle in action. Previously, I wrote about how I planned for a problem-solving lesson and then described my instruction during the following lesson. In this post, I want to discuss how I used assessment and evaluation to monitor the learners' progress and inform future planning and instruction.


We were using the clock model as a context for adding fractions and I wanted to gather data about whether or not the learners could determine when this model was an effective approach. I used an existing set of textbook items and asked the fifth-graders to: "Look at the expressions shown below - circle the ones that you think you could use the clock model to solve and place an 'X' through those you could not."

from Scott Foresman – Addison Wesley Math [5^th Grade]

Once the kids had completed this task, I asked them to solve one of the problems they had circled. As they worked, I gathered data on whether or not they were able to determine when the clock model could work.


In analyzing my observations, I noticed what the fifth-graders could do and what they were trying to do. First, they all recognized that fractions involving ninths and sevenths were poor candidates for the clock model. Those who chose to solve #2, #9, and #10 were also fluent in applying prior experiences with the time context. Some learners thought eighths could work (circling #4) and others struggled to see that fifths could work ('X'ing out #3, #5, and #8). These last two areas of approximation gave me some ideas about what to focus on next.


The last assessment I gave was intended to gather data about how the fifth-graders might apply the idea of context to a problem that could not be easily solved using the clock model. As a ticket out the door, I asked, "Now what could you do to solve a problem you put an 'X' through?" Based on my evaluation of these assessments, I was prepared to plan for future lessons.


What would you do next?

How might this change things?

Here is another statistics workshop used in a class for preservice K-8 math teachers. The goals of this workshop are: (1) develop familiarity with the lists on the TI-84; (2) practice applying Polya's problem-solving phases; and (3) develop conceptual understanding of the measures of center (mean, median, and mode) and variability (mean absolute deviation). It has been updated to reflect my current understandings about teaching and learning and the Common Core State Standards (CCSS).


Grade 7 CCSS in Statistics and Probability [7.SP]
Draw informal comparative inferences about two populations
(3) Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities...
Standards of Mathematical Practice
(1) Make sense of problems and persevere in solving them.


Schema Activation: Teacher Recount
"Last class, we were introduced to the Penny for Peace contest going on between the middle school math classes at a local school. We used this data to develop a procedure for finding the mean absolute deviation (MAD). This involved: (1) finding the mean of the data; (2) finding the distance (absolute value) between the mean and each data point; and (3) averaging these distances to determine the MAD."

Focus: Problem Solving
"Today, we are going to use Polya's problem-solving phases to explore, 'What happens if...' Recall that our modified phases are: Understand the problem; create a plan; carry out your plan; and look back and look forward. Also, remember that the phases are not intended to be linear. Effective problem-solvers jump between the different phases as they work toward a solution. Please be sure to record all of your work and your thinking so we have an artifact that we can refer to during our reflection. Are there any questions?"


Activity: Transforming Data
Understanding the problem

It turns out that the principal, Ms. Sanchez, decided to add $10 to all the class totals.  How do you think this will affect the three central tendencies (mode, median, and mean) and the mean absolute deviation?  Start by making a prediction.

Classes	6.1	6.2	6.3	7.1	7.2	7.3	8.1	8.2	8.3
PfP	$76	$76	$76	$74	$73	$71	$69	$68	$65
PfP+10	$86	$86	$86	$84	$83	$81	$79	$78	$75

Create a plan

With your group, work on a plan to use your calculator's list feature to explore this problem.

Carry out your plan

Keep a record of your efforts and your thinking.

Looking back & looking forward

• What happened to the central tendencies and the MAD when you transformed the data?
• How did you go about obtaining your results?
• Why do you think this happened?
• What do you think would happen if the original amounts were doubled?

Reflection: Problem Solver's Chair

The teacher selects a learner to recount his/her group's problem-solving exhibition. This learner can be a volunteer, selected at random, or chosen based on something interesting observed by the teacher during the activity portion of the workshop. After the recount, other learners offer observations or ideas based on their own experiences

What did you do over summer vacation?

I thought about education, of course.

Dave & Kathy at Mary's Bistro

Mackinac Island is a summer vacation destination located between Michigan's Upper and Lower Peninsulas. The island is home to numerous resorts, a state park, and various historic landmarks. You might recognize its Grand Hotel as the backdrop for the 1980s movie, Somewhere in Time. A unique feature of the island is that motorized vehicles are prohibited. It was in this setting that Kathy and I spent a beautiful day two summers ago taking in the sights as we rode our bikes around the island.

Photo courtesy Todd Van Hoosear

Because no cars or trucks are allowed on the island, horse-drawn vehicles are found everywhere. There are horse taxis, delivery carts, and even street cleaners. During our biking, we happened upon a garbage "truck" being pulled by a team of horses. As we passed, we noticed that the driver was fast asleep. It didn't matter. The horses knew where to go - probably after years of following the same path. I was impressed.

I have been thinking about that experience recently (and not just because summer is once again upon us). Current education reform efforts have me wondering if what policy makers have in mind is a system that trains kids to be like those horses we encountered on Mackinac Island. Seth Godin might call it training cogs. Cogs that can follow a predetermined path while those in charge feel safe enough to take a nap. Probably the most frustrating part is that this is not real reform, just reorganizing a system that already rewards conformity and penalizes creativity. In fact, much like transportation on Mackinac Island, educational reformers seem satisfied using "vehicles" (high-stakes tests, teacher-proof curricula, back-to-basics resources ...) from the past. (Read Linda Darling-Hammond's commencement address at Teachers College for more examples)

In researching this post, I found the following. 

Where Mackinac Island Horses Spend the Winter - Pickford, Michigan
Photo courtesy Kate Ter Harr

This photograph represents some of the ideas I have for education reform: (1) freedom to explore; (2) opportunity to collaborate; and (3) open space that allows for choice. The metaphor is not exact but it doesn't have to be. It's got me thinking. And I cannot imagine a better way to spend my "summer vacation."

What support do learners need?

Last week I shared a lesson plan used to introduce the clock model for adding and subtracting fractions to a class of fifth-graders. This post focuses on the follow-up lesson, which concentrated on developing an anchor chart that learners could lean on as they solved progressively more difficult fraction computation problems. In particular, I will discuss how the instruction attempted to offer support so new learning could occur.

Learning is about moving from the known to the new. Therefore, we began by activating our schema regarding how we had used the clock model to add fractions in the previous lesson. Then we considered other fractions that could be represented using time as a context and began building an anchor chart based on our experiences with clocks in and out of the classroom.

An anchor chart supports learning by creating a record that learners can refer to as cognitive demand increases. In Reading with Meaning, Debbie Miller writes, "Anchor charts make our thinking permanent and visible, and so allow us to make connections from one strategy to another, clarify a point, build on earlier learning, and simply remember a specific lesson (p. 57)." This anchor chart offered the fifth-graders support both as a representation for thinking as they computed the fractions and a representation of thinking that they could point to as they communicated their thinking to their peers.

We worked through the number string together, with me starting the computation and learners offering advice as we went along. A number string is a series of progressively more difficult problems that build on the success of prior solutions. Being sure to use "I language," I started each problem by saying, "This reminds me of ..." As we went on, I asked more and more often, "What should I do next?"


From the perspective of the Gradual Release of Responsibility, my approach for this lesson would be considered Shared Practice [WITH]. (In the previous lesson, I had relied on Demonstration [TO] in order to support the introduction of the clock context - something new. Based on learners' efforts in that lesson, I was confident that they were ready for more responsibility.) With Shared Practice, the teacher supports learners by reinforcing how problem solvers get started, but there is still room for exploration and approximation as learners offer their suggestions for what comes next. This came in the form of the fifth-graders telling me what to do to complete the fraction problems. Not all of their suggestions worked, but we thought through them together and used our prior knowledge and anchor chart to get back on track. Any "mistakes" were used as an opportunity to foster a learning community that could support each other through difficult problems.

At the close of this lesson, I offered a final support - time to reflect. Without an opportunity to consolidate their experiences from the lesson, it is very likely that the learning would not last. I asked the fifth-graders to write in their journals a recount of the day using the What, So What, and Now What framework. Their responses would serve as a formative assessment used to inform future lessons. But that is for another post.